Effects of Pump Photon Energy on Generation and Ultrafast Relaxation of Excitons and Charge Carriers in CdSe Nanoplatelets

We studied the initial nature and relaxation of photoexcited electronic states in CdSe nanoplatelets (NPLs). Ultrafast transient optical absorption (TA) measurements were combined with the theoretical analysis of the formation and decay of excitons, biexcitons, free charge carriers, and trions. In the latter, photons and excitons were treated as bosons and free charge carriers as fermions. The initial quantum yields of heavy-hole (HH) excitons, light-hole (LH) excitons, and charge carriers vary strongly with photon energy, while thermal relaxation occurs always within 1 ps. After that, the population of LH excitons is negligible due to relaxation to HH excitons or decay into free electrons and holes. Up to the highest average number of about four absorbed photons per NPL in our experiments, we found no signatures of the presence of biexcitons or larger complexes. Biexcitons were only observed due to the interaction of a probe-generated exciton with an exciton produced previously by the pump pulse. For higher pump photon energies, the initial presence of more free charge carriers leads to formation of trions by probe photons. On increasing the number of absorbed pump photons in an NPL, the yield of excitons becomes higher as compared to free charge carriers, since electron–hole recombination becomes more likely. In addition to a TA absorption feature at energy below the HH exciton peak, we also observed a TA signal at the high-energy side of this peak, which we attribute to formation of LH-HH biexcitons or trions consisting of a charge and LH exciton.

. Decay kinetics of TA traces normalized at maximum amplitude after excitation with a pump photon energy equal to !! (first column, red), "! (middle column, green) and #$%& (right column, blue). The probe photon energy was taken at the maximum of the PA1 feature (top row), the HH bleach (middle row) and at the high-energy bleach feature around "! = 2.58 eV (bottom row). The probe photon energies varies during about 1 ps after the maximum TA amplitude has been reached, due to the change of the shape of the TA spectra on this timescale, as discussed in the main text. Traces are shown for different average numbers of absorbed pump photons per nanoplatelet, '( , which were determined as described below. The average number of absorbed pump photons with energy )*+) by a NPL is equal to '( ( )*+) ) = , ( )*+) ) ( )*+) ) -." , (S1) with , ( )*+) ) the laser fluence in a pump pulse, ( )*+) ) the absorption coefficient of a NPL with volume -." . The incident fluence was calculated according to , ( )*+) ) = )*+) )*+) )*+) where )*+) is the pump power measured using a pinhole with radius = 1 mm, and )*+) = 2500 Hz is the frequency of pump pulses arriving at the sample. The uncertainty of )*+) is about 0.1 mW, which stems from the laser instability during time and inaccuracy of the overlap between the pump and probe laser pulses in the sample. Values of ( )*+) ) were obtained as described previously. 1 For photoexcitation to the HH exciton at 2.42 eV, the LH exciton at 2.58eV and into the continuum at 3.10 eV this gives, = 3.25 × 10 0 cm 12 , = 2.36 × 10 0 cm 12 and = 1.80 × 10 0 cm 12 , respectively. The volume, -." , of the NPLs was calculated from their lateral sizes ( 3 = 25 ± 3 nm, 4 = 8 ± 1 nm) and their width of 4.5 monolayers ( 5 = 1.37 nm) yielding -." = 3 4 5 = 280 ± 45 nm 6 . 2 Figure S1 shows the measured HH bleach amplitude 2 ps after
The optical absorption signal after the pump pulse is affected by the presence of excitons and charge carriers, which we describe according to with O C , O H and , O I the average number of excitons, free electrons and holes in a NPL, respectively. The factors C and DEFG take into account the enhanced transmitted fluence of probe light due to (probe-pulse induced) photon emission from excitons and Pauli blocking by charge carriers, respectively. Note, that in Equation (S6) we make the approximation that the cross sections depend on the average number of excitons and charge carriers rather than their actual number. As discussed in Section S6, the occurrence of quenching of excitons by charge trapping causes the cross sections of excitons and remaining charge carriers to shift to higher energy. In the absence of photoexcitation of the sample by the pump pulse we have C = H = I = 0, and C = #$%& = 0, so that Equation (S4) reduces to the ground state absorption spectrum $77 (ℏ ) = $77 (ℏ ) (10) = [ C (ℏ ) + DEFG (ℏ )] 89: . Note, that C and DEFG depend on the center-of-mass (COM) motion of excitons, which is determined by the lateral confinement in the NPLs, as discussed in Section S4 and in our previous work. 2 We also take into account that the measured optical absorption, $% (ℏ ), depends on the distribution of excitons and charge carriers over (thermal) energy ( J ), see Sections S3 -S5.

S3. Transient absorption due to excitons
According to the Fermi's golden rule, the rate for an optical transition of an electron between initial (i) and final (f) states is given by 3 where the perturbation K%& describes the dipolar interaction between the electron and the electric field due to photons and C (ℏ ) is the density of exciton states at photon energy ℏ above the ground state. Photons are true bosons, while excitons can to a good approximation be treated as bosons provided their mutual distances are much longer than their Bohr radius. [4][5] Taking into account the bosonic nature of these species, the average rate of photon absorption over all NPLs with initial states \ C , N ], of C excitons and interacting with N probe photons, to a final state | C + 1, N − 1⟩ is In Equation (S8), a c is the photon annihilation operator, Ψ a C O is the exciton creation operator, -' (ℏ ) and -( (ℏ ) are the probabilities of having a NPL with C excitons interacting with N photons of energy ℏ . The probability -' is determined by the product of the number of pump photons absorbed in a NPL (Poisson distribution) and the quantum yield of excitons with energy ℏ . The factor Γ C (ℏ ) brings into account the optical transition dipole moment and density of final states. We have taken this factor out of the summation, which implies that the optical absorption cross section C (ℏ , O C ) is assumed to S5 be determined by the average number ( O C ) of excitons in a NPL, rather than the actual number. Analogous to Equation (S6), the average rate for emission of a photon with an exciton recombining to the ground state is given by The net rate resulting from both photon absorption and emission is then where we have used In Equation (S11) and below we explicitly specify the average number of excitons and photons at a particular probe photon energy ℏ by using the notation O K (ℏ ). In Equation (S10), the last term is due to spontaneous emission only. The stimulated emission is cancelled by the Bose enhancement of the absorption.
The decay of the average number of probe photons O N (ℏ , ) per unit path length at position in the sample is with c the velocity of light in the sample and is the volume encountered by the probe pulse. Since O N (ℏ , ) is directly proportional to the laser probe fluence (ℏ , ), we can use Equations (S10) and (S12) to get with C (ℏ , O C ) = Γ C (ℏ ) / the optical absorption cross section. Note that the average number of excitons, , depends on the position due to absorption of the pump laser pulse along its path through the sample. Integrating Equation (S13) over the optical path length yields and the first term in Equation (S6) due to excitons is then found to be so that The factor C (ℏ ) is determined by the bosonic nature of photons and excitons (i.e. the terms '+1' in Equation (S10)).
The initially photogenerated excitons or free electrons and holes will relax thermally. We assume this leads to formation of HH excitons and charge carriers. We only consider HH excitons, since the population of excitons with higher energy is negligible, as discussed before. 6 For a Bose-Einstein distribution of HH excitons over COM states their chemical potential, C , is an independent fitting parameter. Fortunately, for the size of our NPLs with each containing at most 4 excitons a Maxwell-Botlzmann distribution is a good approximation, see Figs. 1 and 7 in our previous work. 6 For a Maxwell-Boltzmann distribution is proportional to a thermal energy distribution factor 1U(ℏ>1V ' ) , where = 1/ J . Now the factor 1UV ' can be included in the fitted amplitude of C (ℏ ), see eq. S17. During time, excitons decay back to the ground state and consequently O C (ℏ , ) decreases with the time delay between the pump and probe laser pulses. These two effects are taken into account by expressing Equation (S16) as the product of the thermal distribution factor and another factor describing the decay of the total exciton population by photon emission (EM) according to

S7
Comparing Equations (S16) and (S17) shows that C,WX ( ) is inversely proportional to the photon fluence, O N (ℏ , ), of the probe laser pulse. The latter is due to the bosonic nature of the photons and excitons. To the best of our knowledge, the dependence of the transient optical absorption on the probe fluence has not been explicitly taken into consideration before.
Note that Equation (S15) depends on the ratio of the number excitons in a NPL and the number of probe photons arriving in a As mentioned at the end of Section S2, for comparison with experiments we must average Equation (S18) over the occupation of excitons in states with different COM translational energy and bring into account the lateral size distribution of the NPLs, as well as effects of (in)homogenous broadening. This is described next in Section S4.

S4. Effects of exciton COM energy, NPL size distribution and disorder on transient absorption due to excitons
Similar to our previous work, we write the wave function of an exciton as 2 , Note that % $c# ( ) is non-zero only if both 3 and 4 are odd. According to Equation (S21) C in Equation (S18) depends on and . For comparison with experiments, we need to average C (ℏ ) over the occupation of states with different and the lateral size distribution ( ) of the NPLs. In addition, we take into account the homogeneous broadening due to exciton-S8 phonon interaction by a Lorentzian profile with full-width at half maximum (FWHM) equal to Γ a (ℏ ). The inhomogeneous broadening due to disorder is described by a Gaussian with standard deviation given by a . These two contributions to the broadening are described by a Voigt-profile, , centered at the energy of the n-th COM exciton state. This implies that the optical absorption in Equation (S18) needs to be averaged by weighting with 2 Analogous to our earlier work, the transient optical absorption spectrum due to excitons is described by combining Equations (S18) -(S22), leading to 2 Comparison of Equations (S23) and (S18) shows that C (ℏ , O C = 0) = ∑ ∫ ( ) ( ) % C (ℏ , , ). 2 Note, that contributions of internally excited excitons can be neglected, since the oscillator strength is much smaller than for ground state excitons and their population is negligible. 2,6 The values of the parameters b , Γ a (ℏ ) and a were taken from our previous fits to steady-state abosprton and photoluminescence spectra of the same NPLs as studied here. 2

S5. Transient absorption due to charge carriers
We now consider the transient absorption due to holes in the heavy hole valence band (VB) and electrons in the conduction band (CB) with wave vector , and angular momentum projection d / = ±3/2 for holes and d 0 = ±1/2 for electrons.
Transitions from the light hole VB are not taken into account, since these occur at energies above 2.8 eV [see Figure 1(a) of the main text], which is larger than the maximum probe photon energy used in the transient absorption experiments, see Figure   1(b) of the main paper. Below we denote the probability averaged over all NPLs that an electron occupies a state in the VB as e ( , d / ) and for an electron in the CB as # ( , d 0 ). The net rate of photon absorption and emission in the presence of charge carriers, analogous to Equation (S10) for excitons, is After optical excitation by the laser pump pulse, the factor N e ( )(1 − # ( )) describes reduced optical absorption due to removal of an electron from the VB and/or population of an electron state in the CB (Pauli blocking), while # ( )(1 − e ( ))( N + 1) is due to emission of a photon via recombination of an electron and a hole. The prime (′) in the summation over indicates that only electrons with a wave vector such that the energy difference between the CB and VB S9 states equals the probe photon energy ℏ contribute to the Equation (S24). Note that in Equation (S24) Γ #$%& (ℏ ) takes into account the optical oscillator strength. By defining the hole probability I ( ) = 1 − e (− ) we can rewrite Equation (S24) as To connect with the notation in Equations (S12-S15) we recall that the average number of probe photons interacting with a while the number of allowed optical transitions in a NPL at probe photon energy ℏ is with H and I the chemical potential of electrons and holes, respectively. Scattering of electrons and holes between different values occurs on a timescale comparable to or less than the oscillation period of optical phonons (~150 fs). This timescale is within the time resolution of our experiments, which is roughly the sum of the pump and probe pulse durations (~400 fs).
Therefore, we assume that the initial correlation of the k vectors of an electron and a hole produced by absorption of a photon is lost and we can make the approximation Note that we ignore the discreteness of electron and hole states due to the finite lateral deimensions of the NPLs. This is a good approximation, since according to recent theoretical modeling the energetic spacing between the electronic states in CdSE NPLS is a few meV, 8 which is comparible to energetic broading resulting from dephasing by e.g. electron-phonon scattering. 9 In the classical regime the term H is very negative, so that (S37) leads to By defining factors that take into account the decay of charge carriers during time as For fitting to the experimental data, Equation (S40) must be averaged over the NPL size distribution and broadened by inclusion of a Voigt profile (analogous to Equation (S22) for excitons in Section S4). For the combined electron and hole continua we get where Γ p rr (ℏ ) and p rr represent the homogeneous and inhomogeneous broadening of transitions to the continuum, respectively, ( ) is the Heaviside step-function and C !! is the HH exciton binding energy (here defined positive). Averaging Equation (S34) using the relative weights of Equation (S35) yields where #$%& is a constant analogous to b in Equation (S23).

S6. Transient absorption features and due to biexcitons and trions
The absorbance features ( 2 and / ) in the Δ spectra in Figures 2 and 4 of the main text could be described by Gaussians centered energy u ) with standard deviation (i=1,2). We attribute these features to formation of biexcitons and trions by absorption of a probe photon near a HH exciton or a charge carrier produced by the pump pulse, as further discussed below and in the main text. We found from our fits that u ) and depend on time. We attribute this to the time-dependence of the number of excitons and charge carriers.

S7. Total transient absorbance
Taking into account the effects of excitons, charge carriers and biexcitons/trions we find that the total transient absorption signal is given by In what follows we define the probe photon energy by ) = ℏ . Combining Equations (S5), (S23), (S36) and adding the contribution of biexcitons/trions (Section S6) the absorbance after the pump laser pulse is found to be (S45) S13 Note that a time-dependent energy shift, Δ ( ), was needed to describe the presence of a double absorbance feature in $% at early times after HH excitation, and a high-energy shoulder of the HH peak after LH and continuum excitation (see Figure S3).
We attribute this shift due to Coulomb screening by free charge carriers or filling of charge traps (via free charges or exciton quenching). In Equation (S45), the first two terms represent the ground state absorbance, which is equal to , ( ) ) = C ( ) ) + $77 ( ) ). The third term containing the factor − C • ) + Δ ( )describes bleach due to (spontaneous) photon emission from excitons. The fourth and fifth terms with factors − 2 • ) + Δ ( )and − / • ) + Δ ( )take into account bleach due to Pauli blocking by holes in the valence band and electrons in the conduction band, as well as their radiative recombination.
We attribute the sixth term with the Gaussian function 2 • ) , u 6 ( ), 1 ( )to the probe pulse producing a HH exciton near a HH exciton or a charge carrier produced by the pump pulse. The probe pulse then forms a HH-HH biexciton or a trion consisting of a HH exciton and an electron or a hole. The last term with the Gaussian function / • ) , u 7 ( ), 2 ( )is attributed to contributions from the high-energy absorbance tail due to formation of a HH-HH biexciton by the probe pulse, or formation of a LH-HH biexciton.
In conclusion, transient absorbance spectra are modeled using Equation (S45). Analogous to our previous work, 2 the spectral shapes of C , , 2 and / take into account the COM motion of excitons, the lateral size distribution of NPLs and S14 (in)homogenous broadening via C (ℏ , , ) and #$%& (ℏ , , ) in Equations and (S22) and (S43), respectively. The values of the broadening parameters Γ a (ℏ ), a and Γ p a (ℏ ) and p a are taken from our previous study. 2 Note that the spectral shape of K ( ) , , ) is identical to that of K ( ) + Δ ( ), , ) with the latter only being shifted by Δ ( ).  Table S1. Fitting parameters in Equation (S45). The time-dependence of the parameters obtained from fitting to experimental data is shown in Figure S4. S15 Figure S3. Time evolution of $% spectra after excitation at !! (left column, red), "! (middle column, blue) and #$%& (right column, green). Spectra were obtained for similar '(~3 [squares in Figure S1]. After !! excitation, we resolve a double peak at early time. This, as well as the portion of the spectrum between the HH and LH peak at higher excitation energies, can be modeled by taking into account a time-dependent energy shift ΔE due to Coulomb screening and trap filling [see Equation